Lecture 6-cs345-2014
- 1. Design and Analysis of Algorithms (CS345/CS345A) Lecture 6 Augmented BST for • Dynamic sequences • Orthogonal range searching 1
- 2. The fundamental question we answered in last class Question: What makes BST pervasive in the world of data structures ? Answer: Augmentation : “storing extra fields at each node” : Additional information
- 3. DATA STRUCTURES FOR DYNAMIC SEQUENCES 3
- 5. Representing sequence using a BST : size field 8 3 4 1 2 1 1 1
- 6. u 6
- 7. T We need to do the rebalancing of height later on. 7
- 8. Example 2: sequence of numbers 8
- 9. Performing Add(T,i,j,x) View the entire tree T from perspective of the paths from i and j to the root. How will T look like ? LCA(i,j) j i T 9
- 10. Performing Add(T,i,j,x) What are the nodes whose values has to be incremented by x ? LCA(i,j) j i T 10
- 11. Performing Add(T,i,j,x) What are the nodes whose values has to be incremented by x ? LCA(i,j) j i T 11
- 12. Performing Add(T,i,j,x) What are the nodes whose values has to be incremented by x ? LCA(i,j) j i T 12
- 14. Key observations about the data structure 14
- 15. Homework 15
- 16. Example 2: sequence of bits 16
- 17. Example 3: sequence of numbers 17
- 18. ORTHOGONAL RANGE SEARCHING Hopefully you have got a fair amount of understanding of augmenting a BST by now. Think over this problem for a while before proceeding further. 18
- 19. Orthogonal Range searching Rectangle No. of points in query rectangle 19
- 22. LCA T 22
- 23. • Binary search tree on x-coordinates LCA T 23
- 24. • Binary search tree on x-coordinates LCA T 24
- 25. • Binary search tree on x-coordinates LCA T 25
- 26. How should we augment each node? Augment each node v with a pointer to another BST storing all points of T(v) according to y-oordinates. value Color-bit Y-tree left right v BST storing T(v) according to y-coordinate 26
- 27. LCA T 27
- 28. Homework 28
- 29. Space of the data structure 1 2 3 4 ... ... T BST before augmentation 29
- 30. Space of the data structure ... ... T BST before augmentation 30
- 31. Total time to build the data structure p 31